Hi,
I don't know of any specific rules but here are some guidelines to get you started...
"til at"/bare "at"
I think of ...
at + infinitive in Danish as the equivalent of
to+infinitive in English, e.g.
a place to stay (et sted
at være),
a place to live (et sted
at bo),
a way to do it (en måde
at gøre det på),
I can't think of anything to do (jeg kan ikke finde på noget
at gøre),
I have something exciting to tell you (jeg har noget spændende
at fortælle dig)
til at + infinitive frequently expresses a purpose, "meant for",
the spoon is for eating soup (skeen er
til at spise suppe med),
these boots are for hiking (disse støvler er
til at trave i),
fluted glassware is for champagne/for drinking champagne (de høje flûte glas er
til at drikke champagne af),
"for at" (or "således at" (formal), så, sådan så (colloquial))
This corresponds to the English "in order to", "so that", or just "to"...and indicates an intention
I stopped by to say hello (jeg kom forbi for at sige hej/for at hilse på dig),
I ran all the way to the station to catch the train (jeg løb hele vejen til stationen for at (kunne) nå toget),
I mowed the lawn so that we could play soccer (jeg slog græsset for at vi kunne spille fodbold).
Be careful with "false-friend" constructions...
for at in Danish is not used as "for" in English.
This knife is
for carving poultry (denne kniv er
til at skære fjerkræ med. (Not "for at skære fjærkræ med"

)
The knife was just sharpened
so that you can carve the turkey,(kniven er lige blevet slebet
for at du kan skære kalkunen ud med den. (not "til at du kan skære kalkunen ud med den"

)
I suspect there will be exceptions... feel free to post them if you think of any.
PS: Your written Danish is really good, but allow me to make a small correction, (either ~ den ene eller den anden/det ene eller det andet)
...men jeg har endnu ikke fundet nogen regler om, hvornår man bruger den ene eller den anden (i.e. konjunktion).